aa r X i v : . [ m a t h . C V ] M a r HOLOMORPHIC CURVES WHOSE DOMAINS ARERIEMANN SURFACES
XIANJING DONG
Abstract.
We establish a defect relation of holomorphic curves from ageneral open Riemann surface into a normal complex projective variety,with Zariski-dense image intersecting effective Cartier divisors. Introduction
Value distribution of holomorphic curves has grown into a very rich branchin Nevanlinna theory [14, 15] since H. Cartan [5] established his Second MainTheorem of holomorphic curve from C into P n ( C ) intersecting hyperplanes ingeneral position. Many well-known results were obtained, referred to Ahlfors[1], Nochka [11, 12], Noguchi-Winkelmann [13, 14], Ru [15, 16, 17, 18, 19],Shabat [20], Tiba [21] and Yamanoi [22], etc. In the paper, we would furtherdevelop the well-known Ru’s result of holomorphic curves by generalizing thesource space C to a general open Riemann surface through Brownian motioninitiated by Carne [6] and developed by Atsuji [2, 3].Let S be an open Riemann surface. By uniformization theorem, one couldequip S with a complete Hermitian metric ds = 2 gdzdz such that the Gausscurvature K S ≤ g, here K S is defined by K S = −
14 ∆ S log g = − g ∂ log g∂z∂z . Obviously, (
S, g ) is a complete K¨ahler manifold with associated K¨ahler form α = g √− π dz ∧ dz. Set(1) κ ( t ) = min (cid:8) K S ( x ) : x ∈ D ( t ) (cid:9) which is a non-positive and decreasing continuous function defined on [0 , ∞ ) . Fix o ∈ S as a reference point. Denoted by D ( r ) the geodesic disc centeredat o with radius r, and by ∂D ( r ) the boundary of D ( r ) . By Sard’s theorem,
Mathematics Subject Classification.
Key words and phrases.
Holomorphic curve; Algebraic variety; Riemann surface; Defectrelation; Brownian motion.
X.J. DONG ∂D ( r ) is a submanifold of S for almost all r > . Also, we denote by g r ( o, x )the Green function of ∆ S / o, and by dπ ro ( x ) the harmonic measure on ∂D ( r ) with respect to o. Let f : S → X be a holomorphic curve, where X is a complex projective variety. Let us firstintroduce Nevanlinna’s functions on Riemann surfaces which are extensionsof the classical ones on C . Let L → X be an ample holomorphic line bundleequipped with Hermitian metric h. We define the characteristic function of f with respect to L by T f,L ( r ) = π Z D ( r ) g r ( o, x ) f ∗ c ( L, h )= − Z D ( r ) g r ( o, x )∆ S log h ◦ f ( x ) dV ( x ) , where dV ( x ) is the Riemannian volume measure of S. It can be easily knownthat T f,L ( r ) is independent of the choices of metrics on L, up to a boundedterm. Since a holomorphic line bundle on X can be written as the differenceof two ample holomorphic line bundles, the definition of T f,L ( r ) can extendto an arbitrary holomorphic line bundle. For a convenience, we use T f,D ( r )to replace T f,L D ( r ) for an effective Cartier divisor D on X. Given an ampleeffective Cartier divisor D on X, the Weil function of D is well defined by λ D ( x ) = − log k s D ( x ) k up to a bounded term, and here s D is the canonical section associated to D. Note also that an effective Cartier divisor can be written as the difference oftwo ample effective Cartier divisors, and so the definition of Weil functionscan extend to anarbitrary effective Cartier divisor. We define the proximityfunction of f with respect to D by m f ( r, D ) = Z ∂D ( x ) λ D ◦ f ( x ) dπ ro ( x ) . Now write s D = ˜ s D e locally, where e is a local holomorphic frame of ( L D , h ) . The counting function of f with respect to D is defined by N f ( r, D ) = π X x ∈ f ∗ D ∩ D ( r ) g r ( o, x )= π Z D ( r ) g r ( o, x ) dd c (cid:2) log | ˜ s D ◦ f ( x ) | (cid:3) = 14 Z D ( r ) g r ( o, x )∆ S log | ˜ s D ◦ f ( x ) | dV ( x )in the sense of currents. OLOMORPHIC CURVES 3
Remark.
When S = C , the Green function is (log r | z | ) /π and the harmonicmeasure is dθ/ π. By integration by part, we observe that it agrees with theclassical ones.We introduce the concept of Nevanlinna constant proposed by Ru.
Definition 1.1 ([17, 18]) . Let L be a holomorphic line bundle over X, and D be an effective Cartier divisor on X. If X is normal, then we define Nev(
L, D ) = inf k,V,µ dim C Vµ , where “ inf ” is taken over all triples ( k, V, µ ) such that V ⊆ H ( X, kL ) is alinear subspace with dim C V > , and µ > is a number with the property:for each x ∈ Supp D, there exists a basis B x of V such that X s ∈ B x ord E ( s ) ≥ µ ord E ( kD ) for all irreducible components E of D passing through x. If there exists nosuch triples ( k, V, µ ) , one defines Nev(
L, D ) = ∞ . If X is not normal, then Nev(
L, D ) is defined by pulling back to the normalization of X. The main purpose of this paper is to explore the value distribution theoryof holomorphic curves into complex projective varieties by extending sourcespace C to a general open Riemann surface. We prove the following theorem Theorem 1.2.
Let L be a holomorphic line bundle over a normal complexprojective variety X with dim C H ( X, kL ) ≥ for some k > . Let D be aneffective Cartier divisor on X. Let f : S → X be a holomorphic curve withZariski-dense image. Then m f ( r, D ) ≤ Nev(
L, D ) T f,L ( r ) + o (cid:0) T f,L ( r ) (cid:1) + O (cid:16) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where κ is defined by (1) , and “ k ” means that the inequality holds for r > outside a subset of finite Lebesgue measure. The term κ ( r ) r in the above theorem appears from the bending of metricof S. In particular, when S = C , it deduces κ ( r ) ≡ T f,L ( r ) ≥ O (log r )for a holomorphic curve f with Zariski-dense image in X. As a consequence,we recover a result of Ru:
Corollary 1.3 ([17]) . The same conditions are assumed as in Theorem . .Let f : C → X be a holomorphic curve with Zariski-dense image. Then m f ( r, D ) ≤ Nev(
L, D ) T f,L ( r ) + o (cid:0) T f,L ( r ) (cid:1)(cid:13)(cid:13) . Theorem 1.2 implies a defect relation
X.J. DONG
Corollary 1.4.
The same conditions are assumed as in Theorem . . Let f : S → X be a holomorphic curve with Zariski-dense image satisfying lim inf r →∞ κ ( r ) r T f,L ( r ) = 0 . Then δ f ( D ) ≤ Nev(
L, D ) . First Main Theorem
Stochastic formulas.
We would use the stochastic method to study value distribution theory forRiemann surfaces. To start with, we introduce Brownian motion and Dynkinformula [8, 10]. It is known that the Dynkin formula plays a similar role asGreen-Jensen formula [15]. Indeed, the co-area formula is also introduced.Let (
M, g ) be a Riemannian manifold with Laplace-Beltrami operator ∆ M associated to g. For x ∈ M, we denote by B x ( r ) the geodesic ball centered at x with radius r, and denote by S x ( r ) the geodesic sphere centered at x withradius r. By Sard’s theorem, S x ( r ) is a submanifold of M for almost every r > . A Brownian motion X t in M is a heat diffusion process generated by ∆ M with transition density function p ( t, x, y ) which is the minimal positivefundamental solution of the heat equation ∂∂t u ( t, x ) −
12 ∆ M u ( t, x ) = 0 . We denote by P x the law of X t started at x ∈ M and by E x the correspondingexpectation with respect to P x . Let D be a bounded domain with smooth boundary ∂D in M . Fix x ∈ D, we use dπ ∂Dx to denote the harmonic measure on ∂D with respect to x. Thismeasure is a probability measure. Set τ D := inf (cid:8) t > X t D (cid:9) which is a stopping time. Denoted by g D ( x, y ) the Green function of ∆ M / D with a pole at x and Dirichlet boundary condition, namely −
12 ∆
M,y g D ( x, y ) = δ x ( y ) , y ∈ D ; g D ( x, y ) = 0 , y ∈ ∂D, where δ x is the Dirac function. For φ ∈ C ♭ ( D ) (space of bounded continuousfunctions on D ), the co-area formula [4] asserts that E x (cid:20)Z τ D φ ( X t ) dt (cid:21) = Z D g D ( x, y ) φ ( y ) dV ( y ) . OLOMORPHIC CURVES 5
From Proposition 2.8 in [4], we also have the relation of harmonic measuresand hitting times that(2) E x [ ψ ( X τ D )] = Z ∂D ψ ( y ) dπ ∂Dx ( y )for any ψ ∈ C ( D ).Let u ∈ C ♭ ( M ) (space of bounded C -class functions on M ), we have thefamous Itˆo formula (see [2, 8, 9, 10]) u ( X t ) − u ( x ) = B (cid:18)Z t k∇ M u k ( X s ) ds (cid:19) + 12 Z t ∆ M u ( X s ) dt, P x − a.s. where B t is the standard Brownian motion in R and ∇ M is gradient operatoron M . Take expectation of both sides of the above formula, it follows Dynkinformula (see [2, 10]) E x [ u ( X T )] − u ( x ) = 12 E x (cid:20)Z T ∆ M u ( X t ) dt (cid:21) for a stopping time T such that each term makes sense. Remark.
Thanks to expectation “ E x ”, the Dynkin formula, co-area formulaand (2) still work when u, φ or ψ has a pluripolar set of singularities.2.2. First Main Theorem.
Let S be a complete open Riemann surface with K¨ahler form α associatedto Hermitian metric g. Fix o ∈ S, we let X t be the Brownian motion withgenerator ∆ S / o ∈ S. Moreover, set a stopping time τ r = inf (cid:8) t > X t D ( r ) (cid:9) . Let f : S → X be a holomorphic curve into a complex projective variety X. Let L → X bean ample holomorphic line bundle equipped with Hermitian metric h. Applyco-area formula, we have T f,L ( r ) = − E o (cid:20)Z τ r ∆ S log h ◦ f ( X t ) dt (cid:21) . A relation of harmonic measures and hitting times implies that m f ( r, D ) = E o (cid:2) λ D ◦ f ( X τ r ) (cid:3) . We here give the First Main Theorem of a holomorphic curve f : S → M such that f ( o ) Supp D, where D is an effective Cartier divisor on X. ApplyDynkin formula to λ D ◦ f ( x ) , E o (cid:2) λ D ◦ f ( X τ r ) (cid:3) − λ D ◦ f ( o ) = 12 E o (cid:20)Z τ r ∆ S λ D ◦ f ( X t ) dt (cid:21) . X.J. DONG
The first term on the left hand side of the above equality is equal to m f ( r, D ) , and the term on the right hand side equals12 E o (cid:20)Z τ r ∆ S λ D ◦ f ( X t ) dt (cid:21) = 12 Z D ( r ) g r ( o, x )∆ S log 1 k s D ◦ f ( x ) k dV ( x )due to co-area formula. Since k s D k = h | ˜ s D | , where h is a Hermitian metricon L D , we get12 E o (cid:20)Z τ r ∆ S λ D ◦ f ( X t ) dt (cid:21) = − Z D ( r ) g r ( o, x )∆ S log h ◦ f ( x ) dV ( x ) − Z D ( r ) g r ( o, x )∆ S log | ˜ s D ◦ f ( x ) | dV ( x )= T f,D ( r ) − N f ( r, D ) . Therefore, we obtainF. M. T. T f,D ( r ) = m f ( r, D ) + N f ( r, D ) + O (1) . Remark. N f ( r, D ) is of a probabilistic expression N f ( r, D ) = lim λ →∞ λ P o (cid:18) sup ≤ t ≤ τ r log 1 k s D ◦ f ( X t ) k > λ (cid:19) . Logarithmic Derivative Lemma
Let (
S, g ) be a simply-connected and complete open Riemann surface withGauss curvature K S ≤ g. By uniformization theorem, thereexists a nowhere-vanishing holomorphic vector field X on S. Calculus Lemma.
Let κ be defined by (1). As is noted before, κ is a non-positive, decreasingcontinuous function on [0 , ∞ ) . Associate the ordinary differential equation(3) G ′′ ( t ) + κ ( t ) G ( t ) = 0; G (0) = 0 , G ′ (0) = 1 . We compare (3) with y ′′ ( t ) + κ (0) y ( t ) = 0 under the same initial conditions, G can be easily estimated as G ( t ) = t for κ ≡ G ( t ) ≥ t for κ . This implies that(4) G ( r ) ≥ r for r ≥ Z r dtG ( t ) ≤ log r for r ≥ . On the other hand, we rewrite (3) as the formlog ′ G ( t ) · log ′ G ′ ( t ) = − κ ( t ) . OLOMORPHIC CURVES 7
Since G ( t ) ≥ t is increasing, then the decrease and non-positivity of κ implythat for each fixed t, G must satisfy one of the following two inequalitieslog ′ G ( t ) ≤ p − κ ( t ) for t >
0; log ′ G ′ ( t ) ≤ p − κ ( t ) for t ≥ . By virtue of G ( t ) → t → , by integration, G is bounded from above by(5) G ( r ) ≤ r exp (cid:0) r p − κ ( r ) (cid:1) for r ≥ . The main result of this subsection is the following
Theorem 3.1 (Calculus Lemma) . Let k ≥ be a locally integrable functionon S such that it is locally bounded at o ∈ S. Then for any δ > , there isa constant C > independent of k, δ, and a subset E δ ⊆ (1 , ∞ ) of finiteLebesgue measure such that E o (cid:2) k ( X τ r ) (cid:3) ≤ F (ˆ k, κ, δ ) e r √ − κ ( r ) log r πC E o (cid:20)Z τ r k ( X t ) dt (cid:21) holds for r > outside E δ , where κ is defined by (1) and F is defined by F (cid:0) ˆ k, κ, δ (cid:1) = n log + ˆ k ( r ) · log + (cid:16) re r √ − κ ( r ) ˆ k ( r ) (cid:8) log + ˆ k ( r ) (cid:9) δ (cid:17)o δ with ˆ k ( r ) = log rC E o (cid:20)Z τ r k ( X t ) dt (cid:21) . Moreover, we have the estimate log F (ˆ k, κ, δ ) ≤ O (cid:16) log + log E o (cid:20)Z τ r k ( X t ) dt (cid:21) + log + r p − κ ( r ) + log + log r (cid:17) . To prove theorem 3.1, we need some lemmas.
Lemma 3.2 ([3]) . Let η > be a constant. Then there is a constant C > such that for r > η and x ∈ B o ( r ) \ B o ( η ) g r ( o, x ) Z rη dtG ( t ) ≥ C Z rr ( x ) dtG ( t ) holds, where G be defined by (3) . Lemma 3.3 ([15]) . Let T be a strictly positive nondecreasing function of C -class on (0 , ∞ ) . Let γ > be a number such that T ( γ ) ≥ e, and φ be astrictly positive nondecreasing function such that c φ = Z ∞ e tφ ( t ) dt < ∞ . Then, the inequality T ′ ( r ) ≤ T ( r ) φ ( T ( r )) holds for all r ≥ γ outside a subsetof Lebesgue measure not exceeding c φ . In particular, take φ ( t ) = log δ t fora number δ > , then T ′ ( r ) ≤ T ( r ) log δ T ( r ) holds for all r > outside asubset E δ ⊆ (0 , ∞ ) of finite Lebesgue measure. X.J. DONG
Proof of Theorem . Proof.
The argument refers to Atsuji [3]. The simple-connectedness and thenon-positivity of Gauss curvature of S imply the following relation (see [7]) dπ ro ( x ) ≤ πr dσ r ( x ) , where dσ r ( x ) is the induced volume measure on ∂D ( r ) . By Lemma 3.2 and(4), we have E o (cid:20)Z τ r k ( X t ) dt (cid:21) = Z D ( r ) g r ( o, x ) k ( x ) dV ( x )= Z r dt Z ∂D ( t ) g r ( o, x ) k ( x ) dσ t ( x ) ≥ C Z r R rt G − ( s ) ds R r G − ( s ) ds dt Z ∂D ( t ) k ( x ) dσ t ( x ) ≥ C log r Z r dt Z rt dsG ( s ) Z ∂D ( t ) k ( x ) dσ t ( x ) , E o (cid:2) k ( X τ r ) (cid:3) = Z ∂D ( r ) k ( x ) dπ ro ( x ) ≤ πr Z ∂D ( r ) k ( x ) dσ r ( x ) . Hence, E o (cid:20)Z τ r k ( X t ) dt (cid:21) ≥ C log r Z r dt Z rt dsG ( s ) Z ∂D ( o,t ) k ( x ) dσ t ( x ) , E o (cid:2) k ( X τ r ) (cid:3) ≤ πr Z ∂D ( r ) k ( x ) dσ r ( x ) . (6)Set Λ( r ) = Z r dt Z rt dsG ( s ) Z ∂D ( t ) k ( x ) dσ t ( x ) . We conclude that Λ( r ) ≤ log rC E o (cid:20)Z τ r k ( X t ) dt (cid:21) = ˆ k ( r ) . Since Λ ′ ( r ) = 1 G ( r ) Z r dt Z ∂D ( t ) k ( x ) dσ t ( x ) , then it yields from (6) that E o (cid:2) k ( X τ r ) (cid:3) ≤ πr ddr (cid:0) Λ ′ ( r ) G ( r ) (cid:1) . OLOMORPHIC CURVES 9
Using Lemma 3.3 twice with (5), then for any δ > ddr (cid:0) Λ ′ ( r ) G ( r ) (cid:1) ≤ G ( r ) n log + Λ( r ) · log + (cid:16) G ( r )Λ( r ) (cid:8) log + Λ( r ) (cid:9) δ (cid:17) o δ Λ( r ) ≤ re r √ − κ ( r ) n log + ˆ k ( r ) · log + (cid:16) re r √ − κ ( r ) ˆ k ( r ) (cid:8) log + ˆ k ( r ) (cid:9) δ (cid:17)o δ ˆ k ( r )= F (cid:0) ˆ k, κ, δ (cid:1) re r √ − κ ( r ) log rC E o (cid:20)Z τ r k ( X t ) dt (cid:21) holds outside a subset E δ ⊆ (1 , ∞ ) of finite Lebesgue measure. Thus, E o (cid:2) k ( X τ r ) (cid:3) ≤ F (cid:0) ˆ k, κ, δ (cid:1) e r √ − κ ( r ) log r πC E o (cid:20)Z τ r k ( X t ) dt (cid:21) . Hence, we get the desired inequality. Indeed, for r > F (ˆ k, κ, δ ) ≤ O (cid:16) log + log + ˆ k ( r ) + log + r p − κ ( r ) + log + log r (cid:17)(cid:13)(cid:13) and log + ˆ k ( r ) ≤ log E o (cid:20)Z τ r k ( X t ) dt (cid:21) + log + log r + O (1) . We have arrived at the required estimate. (cid:3)
Logarithmic Derivative Lemma.
Let ψ be a meromorphic function on ( S, g ) . The norm of the gradient of ψ is defined by k∇ S ψ k = 1 g (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂z (cid:12)(cid:12)(cid:12)(cid:12) in a local coordinate z. Locally, we write ψ = ψ /ψ , where ψ , ψ are localholomorphic functions without common zeros. Regard ψ as a holomorphicmapping into P ( C ) by x [ ψ ( x ) : ψ ( x )] . We define T ψ ( r ) = 14 Z D ( r ) g r ( o, x )∆ S log (cid:0) | ψ ( x ) | + | ψ ( x ) | (cid:1) dV ( x )and T ( r, ψ ) := m ( r, ψ ) + N ( r, ψ ) with m ( r, ψ ) = Z ∂D ( r ) log + | ψ ( x ) | dπ ro ( x ) ,N ( r, ψ ) = π X x ∈ ψ − ( ∞ ) ∩ D ( r ) g r ( o, x ) . Let i : C ֒ → P ( C ) be an inclusion defined by z [1 : z ] . Via the pull-back by i, we have a (1,1)-form i ∗ ω F S = dd c log(1+ | ζ | ) on C , where ζ := w /w and [ w : w ] is the homogeneous coordinate system of P ( C ) . The characteristicfunction of ψ with respect to i ∗ ω F S is defined byˆ T ψ ( r ) = 14 Z D ( r ) g r ( o, x )∆ S log(1 + | ψ ( x ) | ) dV ( x ) . Clearly, ˆ T ψ ( r ) ≤ T ψ ( r ) . We adopt the spherical distance k· , ·k on P ( C ) , theproximity function of ψ with respect to a ∈ P ( C ) is defined byˆ m ψ ( r, a ) = Z ∂D ( r ) log 1 k ψ ( x ) , a k dπ ro ( x ) . Again, set ˆ N ψ ( r, a ) = π X x ∈ ψ − ( a ) ∩ D ( r ) g r ( o, x ) . Then ˆ T ψ ( r ) = ˆ m ψ ( r, a ) + ˆ N ψ ( r, a ) + O (1) . Note that m ( r, ψ ) = ˆ m ψ ( r, ∞ ) + O (1) , which implies that T ( r, ψ ) = ˆ T ψ ( r ) + O (1) , T (cid:16) r, ψ − a (cid:17) = T ( r, ψ ) + O (1) . Hence, we arrive at(7) T ( r, ψ ) + O (1) = ˆ T ψ ( r ) ≤ T ψ ( r ) + O (1) . We establish the following Logarithmic Derivative Lemma (LDL):
Theorem 3.4 (LDL) . Let ψ be a nonconstant meromorphic function on S. Let X be a nowhere-vanishing holomorphic vector field on S. Then m (cid:16) r, X k ( ψ ) ψ (cid:17) ≤ k T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where κ is defined by (1) . On P ( C ) , we take a singular metricΦ = 1 | ζ | (1 + log | ζ | ) √− π dζ ∧ dζ. A direct computation gives that Z P ( C ) Φ = 1 , πψ ∗ Φ = k∇ S ψ k | ψ | (1 + log | ψ | ) α. Set T ψ ( r, Φ) = 12 π Z D ( r ) g r ( o, x ) k∇ S ψ k | ψ | (1 + log | ψ | ) ( x ) dV ( x ) . OLOMORPHIC CURVES 11
By Fubini’s theorem T ψ ( r, Φ) = Z D ( r ) g r ( o, x ) ψ ∗ Φ α dV ( x )= π Z ζ ∈ P ( C ) Φ X x ∈ ψ − ( ζ ) ∩ D ( r ) g r ( o, x )= Z ζ ∈ P ( C ) N ψ ( r, ζ )Φ ≤ T ( r, ψ ) + O (1) . We get(8) T ψ ( r, Φ) ≤ T ( r, ψ ) + O (1) . Lemma 3.5.
Assume that ψ ( x ) . Then E o (cid:20) log + k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) ≤
12 log T ( r, ψ ) + O (cid:0) log + log T ( r, ψ ) + r p − κ ( r ) + log + log r (cid:1)(cid:13)(cid:13) , where κ is defined by (1) . Proof.
By Jensen’s inequality, it is clear that E o (cid:20) log + k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) ≤ E o (cid:20) log (cid:16) k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:17)(cid:21) ≤ log + E o (cid:20) k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) + O (1) . By Lemma 3.1 and (8)log + E o (cid:20) k∇ S ψ k | ψ | (1 + log | ψ | ) ( X τ r ) (cid:21) ≤ log + E o (cid:20)Z τ r k∇ M ψ k | ψ | (1 + log | ψ | ) ( X t ) dt (cid:21) + log F (ˆ k, κ, δ ) e r √ − κ ( r ) log r πC ≤ log T ψ ( r, Φ) + log F (ˆ k, κ, δ ) + r p − κ ( r ) + log + log r + O (1) ≤ log T ( r, ψ ) + O (cid:16) log + log + ˆ k ( r ) + r p − κ ( r ) + log + log r (cid:17)(cid:13)(cid:13) , where ˆ k ( r ) = log rC E o (cid:20)Z τ r k∇ S ψ k | ψ | (1 + log | ψ | ) ( X t ) dt (cid:21) . Indeed, we note thatˆ k ( r ) = 2 π log rC T ψ ( r, Φ) ≤ π log rC T ( r, ψ ) . Then we have the desired inequality. (cid:3)
We first prove LDL for the first-order derivative:
Theorem 3.6 (LDL) . Let ψ be a nonconstant meromorphic function on S. Let X be a nowhere-vanishing holomorphic vector filed on S. Then m (cid:16) r, X ( ψ ) ψ (cid:17) ≤
32 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where κ is defined by (1) . Proof.
Write X = a ∂∂z , then k X k = g | a | . We have m (cid:16) r, X ( ψ ) ψ (cid:17) = Z ∂D ( r ) log + | X ( ψ ) || ψ | ( x ) dπ ro ( x ) ≤ Z ∂D ( r ) log + | X ( ψ ) | k X k | ψ | (1 + log | ψ | ) ( x ) dπ ro ( x )+ 12 Z ∂D ( r ) log(1 + log | ψ ( x ) | ) dπ ro ( x ) + 12 Z ∂D ( r ) log + k X x k dπ ro ( x ):= A + B + C. We next handle
A, B, C respectively. For A, it yields from Lemma 3.5 that A = 12 Z ∂D ( r ) log + | a | (cid:12)(cid:12)(cid:12) ∂ψ∂z (cid:12)(cid:12)(cid:12) g | a | | ψ | (1 + log | ψ | ) ( x ) dπ ro ( x )= 12 Z ∂D ( r ) log + k∇ S ψ k | ψ | (1 + log | ψ | ) ( x ) dπ ro ( x ) ≤
12 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) + r p − κ ( r ) + log + log r (cid:17)(cid:13)(cid:13) . For B, the Jensen’s inequality implies that B ≤ Z ∂D ( r ) log (cid:16) + | ψ ( x ) | + log + | ψ ( x ) | (cid:17) dπ ro ( x ) ≤ log Z ∂D ( r ) (cid:16) + | ψ ( x ) | + log + | ψ ( x ) | (cid:17) dπ ro ( x ) ≤ log T ( r, ψ ) + O (1) . Finally, we estimate C. By the condition, k X k > . Since S is non-positivelycurved and a is holomorphic, then log k X k is subharmonic, i.e., ∆ S log k X k ≥ . Clearly, we have ∆ S log + k X k ≤ ∆ S log k X k OLOMORPHIC CURVES 13 for x ∈ S satisfying k X x k 6 = 1 . Notice that log + k X x k = 0 for x ∈ S satisfying k X x k ≤ . Thus, by Dynkin formula we have C = 12 E o (cid:2) log + k X ( X τ r ) k (cid:3) (9) ≤ E o (cid:20)Z τ r ∆ S log k X ( X t ) k dt (cid:21) + O (1)= 14 E o (cid:20)Z τ r ∆ S log g ( X t ) dt (cid:21) + 14 E o (cid:20)Z τ r ∆ S log | a ( X t ) | dt (cid:21) + O (1)= − E o (cid:20)Z τ r K S ( X t ) dt (cid:21) + O (1) ≤ − κ ( r ) E o (cid:2) τ r (cid:3) + O (1) , where we use the fact K S = − (∆ S log g ) / . Thus, we prove the theorem byusing E o [ τ r ] ≤ r / (cid:3) Lemma 3.7.
Let X t be a Brownian motion in a simply-connected completeRiemann surface S of non-positive Gauss curvature. Then E o (cid:2) τ r (cid:3) ≤ r . Proof.
We refer to arguments of Atsuji [3]. Apply Itˆo formula to r ( x )(10) r ( X t ) = B t − L t + 12 Z t ∆ S r ( X s ) ds, where B t is the standard Brownian motion in R , L t is the local time on cutlocus of o, an increasing process which increases only at cut loci of o. Since S is simply connected and non-positively curved, then∆ S r ( x ) ≥ r ( x ) , L t ≡ . By (10), we arrive at r ( X t ) ≥ B t + 12 Z t dsr ( X s ) . Associate the stochastic differential equation dW t = dB t + 12 dtW t , W = 0 , where B t is the standard Brownian motion in R , and W t is the 2-dimensionalBessel process defined as the Euclidean norm of Brownian motion in R . Bythe standard comparison arguments of stochastic differential equations, onegets that(11) W t ≤ r ( X t ) almost surely. Set ι r = inf (cid:8) t > W t ≥ r (cid:9) , which is a stopping time. From (11), we can verify that ι r ≥ τ r . This implies(12) E o [ ι r ] ≥ E o [ τ r ] . Since W t is the Euclidean norm of the Brownian motion in R starting fromthe origin, then applying Dynkin formula to W t we have E o [ W ι r ] = 12 E o (cid:20)Z ι r ∆ R W t dt (cid:21) = 2 E o [ ι r ] , where ∆ R is the Laplace operator on R . Using (11) and (12), we concludethat r = E o [ r ] = 2 E o [ ι r ] ≥ E o [ τ r ] . This certifies the assertion. (cid:3)
Proof of Theorem . Proof.
Note that m (cid:16) r, X k ( ψ ) ψ (cid:17) ≤ k X j =1 m (cid:16) r, X j ( ψ ) X j − ( ψ ) (cid:17) . Therefore, we finish the proof by using Lemma 3.8 below. (cid:3)
Lemma 3.8.
We have m (cid:16) r, X k +1 ( ψ ) X k ( ψ ) (cid:17) ≤
32 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where κ is defined by (1) . Proof.
We first claim that T (cid:0) r, X k ( ψ ) (cid:1) ≤ k T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . (13)By virtue of Theorem 3.6, when k = 1 T ( r, X ( ψ )) = m ( r, X ( ψ )) + N ( r, X ( ψ )) ≤ m ( r, ψ ) + 2 N ( r, ψ ) + m (cid:16) r, X ( ψ ) ψ (cid:17) ≤ T ( r, ψ ) + m (cid:16) r, X ( ψ ) ψ (cid:17) ≤ T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) holds for r > k ≤ n − . By induction, we only need to prove the claim
OLOMORPHIC CURVES 15 in the case when k = n. By the claim for k = 1 proved above and Theorem3.6 repeatedly, we have T (cid:0) r, X n ( ψ ) (cid:1) ≤ T (cid:0) r, X n − ( ψ ) (cid:1) + O (cid:16) log T (cid:0) r, X n − ( ψ ) (cid:1) − κ ( r ) r + log + log r (cid:17) ≤ n T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) + O (cid:16) log T (cid:0) r, X n − ( ψ ) (cid:1) − κ ( r ) r + log + log r (cid:17) ≤ n T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17) + O (cid:0) log T (cid:0) r, X n − ( ψ ) (cid:1)(cid:1) · · · · · · · · ·≤ n T ( r, ψ ) + O (cid:16) log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . The claim (13) is proved. Using Theorem 3.6 and (13) to get m (cid:18) r, X k +1 ( ψ ) X k ( ψ ) (cid:19) ≤
32 log T (cid:0) r, X k ( ψ ) (cid:1) − κ ( r ) r G ( r ) r + O (cid:16) log + log T (cid:0) r, X k ( ψ ) (cid:1) + log + log G ( r ) (cid:17) ≤
32 log T ( r, ψ ) + O (cid:16) log + log T ( r, ψ ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . (cid:3) Second Main Theorem
Wronskian determinants.
Let S be an open Riemann surface with a nowhere-vanishing holomorphicvector field X (it always exists), which is equipped with a complete Hermitianmetric h such that the Gauss curvature K S ≤ . Let f : S → P n ( C )be a holomorphic curve into complex projective space with the Fubini-Studyform ω F S . Locally, we may write f = [ f : · · · : f n ] , a reduced representation,i.e., f = w ◦ f, · · · are local holomorphic functions without common zeros,where w = [ w : · · · : w n ] denotes homogenous coordinate system of P n ( C ) . Set k f k = | f | + · · · + | f n | . Notice that ∆ S log k f k is independent of thechoices of representations of f, so it is well defined on S. The height functionof f is defined by T f ( r ) = π Z D ( r ) g r ( o, x ) f ∗ ω F S = 14 Z D ( r ) g r ( o, x )∆ S log k f ( x ) k dV ( x ) . Let H be a hyperplane of P n ( C ) with defining function ˆ H ( w ) = a w + · · · + a n w n . Set k ˆ H k = | a | + · · · + | a n | . The counting function of f with respectto H is defined by N f ( r, H ) = π Z D ( r ) g r ( o, x ) dd c (cid:2) log | ˆ H ◦ f | (cid:3) = 14 Z D ( r ) g r ( o, x )∆ S log | ˆ H ◦ f | dV ( x )in the sense of currents. We define the proximity function of f with respectto H by m f ( r, H ) = Z ∂D ( r ) log k ˆ H kk f ( x ) k| ˆ H ◦ f ( x ) | dπ ro ( x ) . Lemma 4.1.
Assume that f k for some k. We have max ≤ j ≤ n T (cid:16) r, f j f k (cid:17) ≤ T f ( r ) + O (1) . Proof.
By (7), we arrive at T (cid:16) r, f j f k (cid:17) ≤ T f j /f k ( r ) + O (1) ≤ Z D ( r ) g r ( o, x )∆ S log (cid:16) n X j =0 | f j ( x ) | (cid:17) dV ( x ) + O (1)= T f ( r ) + O (1) . (cid:3) Let H , · · · , H q be q hyperplanes of P n ( C ) in N -subgeneral position withdefining functions given byˆ H j ( w ) = n X k =0 a jk w k , ≤ j ≤ q. One defines Wronskian determinant and logarithmic Wronskian determinantof f with respect to X respectively by W X ( f , · · · , f n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f · · · f n X ( f ) · · · X ( f n )... ... ... X n ( f ) · · · X n ( f n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ∆ X ( f , · · · , f n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · X ( f ) f · · · X ( f n ) f n ... ... ... X n ( f ) f · · · X n ( f n ) f n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . OLOMORPHIC CURVES 17
For a ( n + 1) × ( n + 1)-matrix A and a nonzero meromorphic function φ on S, we can check the following basic properties:∆ X ( φf , · · · , φf n ) = ∆ X ( f , · · · , f n ) ,W X ( φf , · · · , φf n ) = φ n +1 W X ( f , · · · , f n ) ,W X (cid:0) ( f , · · · , f n ) A (cid:1) = det( A ) W X ( f , · · · , f n ) ,W X ( f , · · · , f n ) = (cid:16) n Y j =0 f j (cid:17) ∆ X ( f , · · · , f n ) . Obviously, ∆ X ( f , · · · , f n ) is globally well defined on S. Lemma 4.2.
Let Q ⊆ { , · · · , q } with | Q | = n + 1 . If S is simply connected,then we have m (cid:16) r, ∆ X (cid:0) ˆ H k ◦ f, k ∈ Q (cid:1)(cid:17) ≤ O (cid:16) log T f ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where κ is defined by (1) . Proof.
We write Q = { j , · · · , j n } and suppose that ˆ H j ◦ f X (cid:0) ˆ H j ◦ f, · · · , ˆ H j n ◦ f (cid:1) = ∆ X , ˆ H j ◦ f ˆ H j ◦ f , · · · , ˆ H j n ◦ f ˆ H j ◦ f ! . Since ˆ H j ◦ f, · · · , ˆ H j n ◦ f are linear forms of f , · · · , f n , by Theorem 3.4 andLemma 4.1 we have m (cid:16) r, ∆ X (cid:0) ˆ H k ◦ f, k ∈ Q (cid:1)(cid:17) ≤ O (cid:16) log T f ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . We have arrived at the desired inequality. (cid:3)
Lemma 4.3.
Let H · · · , H q be hyperplanes of P n ( C ) . Let f : S → P n ( C ) be a linearly non-degenerate holomorphic curve. Then Z ∂D ( r ) max Q X k ∈ Q log k ˆ H k kk f ( x ) k| ˆ H k ◦ f ( x ) | dπ ro ( x ) ≤ ( n + 1) T f ( r ) + O (cid:16) log T f ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where Q ranges over all subsets of { , · · · , q } such that { ˆ H k } k ∈ Q are linearlyindependent. Proof.
Without loss of generality, we assume that q ≥ n + 1 and H · · · , H q are in general position. Then Z ∂D ( r ) max Q X k ∈ Q log k ˆ H k kk f ( x ) k| ˆ H k ◦ f ( x ) | dπ ro ( x )= Z ∂D ( r ) max | Q | = n +1 log Y k ∈ Q k ˆ H k kk f ( x ) k| ˆ H k ◦ f ( x ) | dπ ro ( x ) ≤ Z ∂D ( r ) max | Q | = n +1 log k f ( x ) k n +1 Q k ∈ Q | ˆ H k ◦ f ( x ) | dπ ro ( x ) + O (1)= Z ∂D ( r ) max | Q | = n +1 log (cid:12)(cid:12) ∆ X (cid:0) ˆ H k ◦ f ( x ) , k ∈ Q (cid:1)(cid:12)(cid:12) k f ( x ) k n +1 (cid:12)(cid:12) W X (cid:0) ˆ H k ◦ f ( x ) , k ∈ Q (cid:1)(cid:12)(cid:12) dπ ro ( x ) + O (1) . By W X ( ˆ H k ◦ f, k ∈ Q ) = b Q W X ( f , · · · , f n ) (with | Q | = n + 1) for a nonzeroconstant b Q depending on Q, we further conclude that Z ∂D ( r ) max Q X k ∈ Q log k ˆ H k kk f ( x ) k| ˆ H k ◦ f ( x ) | dπ ro ( x ) ≤ Z ∂D ( r ) max | Q | = n +1 log (cid:12)(cid:12)(cid:12) ∆ X (cid:0) ˆ H k ◦ f ( x ) , k ∈ Q (cid:1)(cid:12)(cid:12)(cid:12) dπ ro ( x )+ Z ∂D ( r ) log k f ( x ) k n +1 (cid:12)(cid:12) W X (cid:0) f ( x ) , · · · , f n ( x ) (cid:1)(cid:12)(cid:12) dπ ro ( x ) + O (1):= A + B + O (1) . We next handle the terms A and B. By Lemma 4.2 A ≤ Z ∂D ( r ) log X | Q | = n +1 (cid:12)(cid:12)(cid:12) ∆ X (cid:0) ˆ H k ◦ f ( x ) , k ∈ Q (cid:1)(cid:12)(cid:12)(cid:12) dπ ro ( x ) ≤ X | Q | = n +1 m (cid:16) r, ∆ X (cid:0) ˆ H k ◦ f, k ∈ Q (cid:1)(cid:17) + O (1) ≤ O (cid:16) log T f ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . Apply Dynkin formula to
B,B = 12 Z D ( r ) g r ( o, x )∆ S log k f ( x ) k n +1 (cid:12)(cid:12) W X (cid:0) f ( x ) , · · · , f n ( x ) (cid:1)(cid:12)(cid:12) dV ( x ) + O (1)= ( n + 1) T f ( r ) − N W X ( r,
0) + O (1) ≤ ( n + 1) T f ( r ) + O (1) . Putting together the above, we have the desired inequality. (cid:3)
Theorem 4.4.
Let D be an effective Cartier divisor on a complex projectivevariety M. Let s , · · · , s q be nonzero elements of a nonzero linear subspace OLOMORPHIC CURVES 19 V ⊆ H ( M, L D ) . Let f : S → M be a holomorphic curve with Zariski denseimage. Then Z ∂D ( r ) max Q X k ∈ Q λ s k ◦ f ( x ) dπ ro ( x ) ≤ dim C V T f,D ( r ) + O (cid:16) log T f,D ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) , where λ s k denotes the Weil function of ( s k ) , and Q ranges over all subsetsof { , · · · , q } such that { s k } k ∈ Q are linearly independent.Proof. Set d = dim C V. If d = 1 , then | Q | = 1 . Hence, for 1 ≤ j ≤ q, we have s k = b k s D for some constant b k = 0 . By the First Main Theorem Z ∂D ( r ) max Q X k ∈ Q λ s k ◦ f ( x ) dπ ro ( x ) ≤ Z ∂D ( r ) λ D ◦ f ( x ) dπ ro ( x ) + O (1) ≤ T f,D ( r ) + O (1) . If d > , we treat the projective space P ( V ) of V that can be regarded as P d − ( C ) . Let M ′ be the closure of graph of f, then there are the canonicalprojection morphisms π : M ′ → M and φ : M ′ → P d − ( C ) . We lift f to˜ f : S → M ′ . Note that (see [18]) there exists an effective Cartier divisor B on M ′ such that for each s ∈ V, we can choose a hyperplane H s (dependingon s ) of P d − ( C ) which satisfies π ∗ ( s ) − B = φ ∗ H s (more precisely, φ ∗ O (1) ∼ = L π ∗ D − B ). For 1 ≤ j ≤ q, one chooses hyperplanes H j of P d − ( C ) such that π ∗ ( s j ) − B = φ ∗ H j . Since M is compact, then we have(14) λ π ∗ ( s j ) = λ φ ∗ H j + λ B + O (1) . In further, we have N ˜ f (cid:0) r, π ∗ ( s j ) (cid:1) = N ˜ f (cid:0) r, φ ∗ H j (cid:1) + N ˜ f ( r, B ) , (15) m ˜ f (cid:0) r, π ∗ ( s j ) (cid:1) = m ˜ f (cid:0) r, φ ∗ H j (cid:1) + m ˜ f ( r, B ) + O (1) . (16)Note that φ ◦ ˜ f : S → P d − ( C ) is a holomorphic curve, using the First MainTheorem, it yields that T φ ◦ ˜ f ( r ) = m φ ◦ ˜ f ( r, H j ) + N φ ◦ ˜ f ( r, H j ) + O (1) . Indeed, L ( s j ) ∼ = L D and f = π ◦ ˜ f are noted. By (15) and (16), we arrive at(17) T f,D ( r ) = T φ ◦ ˜ f ( r ) + T ˜ f,B ( r ) + O (1) . Combining (14) with λ s j ◦ f = λ π ∗ ( s j ) ◦ ˜ f + O (1) , it suffices to show that Z ∂D ( r ) max Q X k ∈ Q (cid:16) λ H k ◦ φ ◦ ˜ f ( x ) + λ B ◦ ˜ f ( x ) (cid:17) dπ ro ( x ) ≤ dT f,D ( r ) + O (cid:16) log T f,D ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . In fact, by Lemma 4.3 and (17) we have Z ∂D ( r ) max Q X k ∈ Q λ H k ◦ φ ◦ ˜ f ( x ) dπ ro ( x )= Z ∂D ( r ) max Q X k ∈ Q log k ˆ H k kk φ ◦ ˜ f ( x ) k| ˆ H k ◦ φ ◦ ˜ f ( x ) | dπ ro ( x ) + O (1) ≤ dT φ ◦ ˜ f ( r ) + O (cid:16) log T φ ◦ ˜ f ( r ) − κ ( r ) r + log + log r (cid:17) ≤ d (cid:0) T f,D ( r ) − T ˜ f,B ( r ) (cid:1) + O (cid:16) log T f,D ( r ) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . Since | Q | ≤ d, the First Main Theorem implies that Z ∂D ( r ) max Q X k ∈ Q λ B ◦ ˜ f ( x ) dπ ro ( x ) ≤ dT ˜ f,B ( r ) + O (1) . Combining the above, we conclude the proof. (cid:3)
Second Main Theorem.
In this subsection, we aim to prove the main theorem of the paper, namely,the Second Main Theorem (Theorem 1.2).Let S be a complete open Riemann surface with Gauss curvature K S ≤ . We here consider the universal covering π : ˜ S → S. By the pull-back of π, ˜ S could be equipped with the induced metric from the metric of S. In such case,˜ S is a simply-connected and complete open Riemann surface of non-positiveGauss curvature. Take a diffusion process ˜ X t in ˜ S so that X t = π ( ˜ X t ) , then˜ X t becomes a Brownian motion with generator ∆ ˜ S / X t start from ˜ o ∈ ˜ S with o = π (˜ o ) , then we have E o [ φ ( X t )] = E ˜ o (cid:2) φ ◦ π ( ˜ X t ) (cid:3) for φ ∈ C ♭ ( S ) . Set ˜ τ r = inf (cid:8) t > X t ˜ D ( r ) (cid:9) , where ˜ D ( r ) is a geodesic disc centered at ˜ o with radius r in ˜ S. If necessary,one can extend the filtration in probability space where ( X t , P o ) are definedso that ˜ τ r is a stopping time with respect to a filtration where the stochasticcalculus of X t works. By the above arguments, we would assume S is simplyconnected by lifting f to the covering. Proof of Theorem . Proof.
Let P be the set of all prime divisors occurring in D, then D = X E ∈ P ord E ( D ) · E. OLOMORPHIC CURVES 21
Set Λ = { σ ⊆ P : ∩ E ∈ σ E = ∅} which is a finite set. For any σ ∈ Λ , we write D = D σ, + D σ, , where D σ, = X E ∈ σ ord E ( D ) · E, D σ, = X E σ ord E ( D ) · E. From the definition of Nev(
L, D ) , for each σ ∈ Λ , there exists a basis B σ ofa linear subspace V k ⊆ H ( X, kL ) with dim C V k > k ) such that1 µ k X s ∈ B σ ord E ( s ) ≥ ord E ( kD )at some (and hence all) points x ∈ ∩ E ∈ σ E. For each E ∈ σ, we have(18) 1 µ k X s ∈ B σ ord E ( s ) · λ E ≥ ord E ( kD ) · λ E . Note that (refer to the proof of Proposition 3.1 in [17]) there exists a number
B > x ∈ X, one can pick σ x ∈ Λ (depending on x ) suchthat λ D σx, ( x ) ≤ B, here B is independent of x . Thus,(19) λ D ( x ) ≤ λ D σx, ( x ) + O (1) . By properties of Weil functions, we have from (19) and (18) that λ kD ( x ) ≤ µ k max σ ∈ Λ X s ∈ B σ λ s ( x ) + O (1) , where λ s ( x ) is the Weil function of ( s ) . Taking the expectation to get km f ( r, D ) ≤ µ k Z ∂D ( r ) max σ ∈ Λ X s ∈ B σ λ s ( x ) dπ ro ( x ) + O (1) . Making use of Theorem 4.4, we obtain km f ( r, D ) ≤ dim C V k µ k T f,kL ( r ) + o (cid:0) T f,kL ( r ) (cid:1) + O (cid:16) − κ ( r ) r + log + log r (cid:17)(cid:13)(cid:13) . This proves Theorem 1.2. (cid:3)
Acknowledgement.
The author is grateful to Prof. Min Ru and Dr. YanHe for their valuable suggestions on this paper.
References [1] Ahlfors L.V.: The theory of meromorphic curves, Acta. Soc. Sci. Fenn. Nova Ser. A (1941), 1-31.[2] Atsuji A.: A second main theorem of Nevanlinna theory for meromorphic functionson complete K¨ahler manifolds, J. Math. Japan Soc. (2008), 471-493.[3] Atsuji A.: Nevanlinna-type theorems for meromorphic functions on non-positivelycurved K¨ahler manifolds, Forum Math. (2018), 171-189. [4] Bass R.F.: Probabilistic Techniques in Analysis, Springer, New York, (1995).[5] Cartan H.: Sur les z´eros des combinaisons lin´eaires de p fonctions holomorphesdonn´ees, Mathematica, (1933), 5-31.[6] Carne T.K.: Brownian motion and Nevanlinna theory, Proc. London Math. Soc. (3) (1986), 349-368.[7] Debiard A., Gaveau B. and Mazet E.: Theorems de comparaison en geometrie Rie-mannienne, Publ. Res. Inst. Math. Sci. Kyoto, (1976), 390-425. (2019), 303-314.[8] Hsu E.P.: Stochastic Analysis on Manifolds, Grad. Stud. Math. 38, American Math-ematical Society, Providence, (2002).[9] Ikeda N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes,2nd edn. North-Holland Mathematical Library, Vol. 24. North-Holland, Amsterdam,(1989).[10] Itˆo K. and McKean Jr H P.: Diffusion Processes and Their sample Paths, AcademicPress, New York, (1965).[11] Nochka E.I.: Defect relations for meromorphic curves (Russian), Izv. Akad. NaukMold. SSR, Ser. Fiz.-Teh. Mat. Nauk (1982), 41-47.[12] Nochka E.I.: On the theory of meromorphic functions, Sov. Math. Dokl. (1983),377-381.[13] Noguchi J. and Winkelmann J.: A note on jets of entire curves in semi-abelian vari-eties, Math. Z. (2003), 705-710.[14] Noguchi J. and Winkelmann J.: Nevanlinna theory in several complex variablesand Diophantine approximation, A series of comprehensive studies in mathematics,Springer, (2014).[15] Ru M.: Nevanlinna theory and its relation to diophantine approximation, WorldScientific Publishing, (2001).[16] Ru M.: Holomorphic curves into algebraic varieties. Ann. of Math. (2009), 255-267.[17] Ru M.: A defect relation for holomorphic curves intersecting general divisors in pro-jective varieties. J. Geom. Anal., (4), (2016), 2751-2776.[18] Ru M.: A Cartan’s Second Main Theorem Approach in Nevanlinna Theory, ActaMathematica Sinica, English Series, (2018), 1-17.[19] Ru M. and Sibony N.: The second main theorem in the hyperbolic case, Mathema-tische Annalen, (2020), 759-795.[20] Shabat B.V.: Distribution of Values of Holomorphic Mappings, Translations of Math-ematical Monographs, (1985).[21] Tiba Y.: Holomorphic curves into the product space of the Riemann spheres, J. Math.Sci. Univ. Tokyo (2011), 325-354.[22] Yamanoi K.: Holomorphic curves in abelian varieties and intersections with highercodimensional subvarieties, Forum Math. (2004), 749-788. Academy of Mathematics and Systems Sciences, Chinese Academy of Sci-ences, Beijing, 100190, P.R. China
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